Enhancing noise-induced escape in systems with distributed delays

Many real world dynamical systems exhibit complex behavior often induced by intrinsic time delays, as well as influenced by random perturbations. An important problem, therefore, is to understand how random disturbances are organized such that the dynamics escape from a stable
attractor and exhibit new behavior. Although the noise amplitudes are small, the result is a large fluctuation out of the basin of attraction. Here we study the influence of noise-induced
large fluctuations on dynamical systems, where the time delay is not taken as a constant but is rather chosen from a given distribution. We use a variational approach to calculate the escape rates out of
the basin of attraction of the stable equilibrium for a general kernel of the delay distribution, as well as general nonlinearity. We illustrate the theory by taking two particular examples of the distribution kernel, namely, the uniform and bi-modal kernels, and analyze how the width of the distribution affects the switching rates. Our results show that the switching is affected not only by the mean time delay but also by the width of the delay distribution. Specifically, if the distribution width is increased, this leads to an increase in the switching times.